# Quantum coherence - what is it's definition?

I stumbled across several definitions of "coherence(s)":

• Pade (Hyperlink) just calls the off-diagonal terms in the density-matrix "coherences"

• Schlosshauer (Hyperlink) says its a measure for the "quantumness" of a system and the reason for it's quantum-effects, doesn't go into detail though.

• Wikipedia (Hyperlink) even refers to work that shows that coherence is equivalent to quantum entanglement.

Since "coherence" seems to be (from by perspective, which just tries to clarify where the name decoherence comes from) a dispensable notation for the property of quantum states to be in superpositions (just like potential energy is a notation for the property of a bike to e.g. accelerate down a slope) I would like to ask:

Is "quantum coherence" just another term for the property of a system to be in superpositions of states?

Coherence in Superpositions

Coherences are the phase relationships between sections of our system.

As you noted, coherence is related to the superposition of states $$|\psi\rangle = w_a |a\rangle + w_b e^{i\phi} |b\rangle$$: if $$\phi$$ is well defined, then there is perfect coherence between $$|a\rangle$$ and $$|b\rangle$$. As Anna states, this proves to have crucial experimental consequences for interference of probabilities. If you add another state $$|\psi'\rangle = e^{i\theta}( w'_a |a\rangle + w'_b e^{i\phi'} |b\rangle)$$ in superposition with $$|\psi\rangle$$ to form $$|\psi\rangle+|\psi'\rangle$$, then it is clear the probabilities (to be measured in a particular basis state) depend on not only the relative phases within both $$|\psi\rangle$$ and $$|\psi'\rangle$$ but also the relative phase $$\theta$$ between them.

If the phase relationship is not there, the probability to be measured in a state adds classically, yielding $$p_a=|w_a|^2+|w'_a|^2$$ for example, with no phase dependence (i.e. interference) to be seen.

As Schlosshauer and Wikipedia either mention or allude, behaving classically (i.e. zero coherence between states) means we also lose quantum effects like quantum entanglement.

Coherences in the Density Matrix

The density matrix $$\rho$$ offers a nice way to smoothly transition between pure states with perfect coherence, and "mixed states" with less-than-perfect coherence between internal states. Any mixed state can be seen as a probabilistic sum of pure states (that by definition have perfect coherence).

If the magnitude of a coherence (an off-diagonal element $$\rho_{ij}$$ of the density matrix) is smaller than it would be for a pure state of the same populations (diagonal elements $$\rho_{ii}$$), then we know the state that $$\rho$$ represents is mixed and cannot fully be interfered. In model, this is because the phase information in the coherences have partially canceled in our probabilistic sum, leading to phase uncertainty.

If all of the coherences $$\rho_{ij}$$ are zero, then the mixed state is fully decohered and acts classically.

This justifies calling an off-diagonal element $$\rho_{ij}$$ a coherence, because its magnitude represents the strength of the phase relationship between the states, and the phase of one is precisely the "averaged" phase relationship of the mixtures.

• Thank you! Can you recommend any definition of coherence that I could cite in a book? Commented May 3, 2022 at 13:09
• Doing a quick search, I found most sources either are focused on rigorous definition (e.g. within a resource theory) or just mention the concept and give a description rather than a useful definition with context. Karnieli et al. (2021) have a quick quotable summary hiding in paragraph 6, but unfortunately don't try to explain interference, or classical vs. quantum interference phenomena. Commented May 3, 2022 at 14:31
• But as long as you make that connection to interference, you could follow up with e.g. Feynman's classic exposition of how interference of probability (amplitudes) via superposition is at the heart of quantum mechanics. Commented May 3, 2022 at 14:42
• @JonathanJeffrey The thing I can't figure out is, what kind of process leads to the summing of states as in your example? A simple example of decoherence is to see that (dropping normalizing factors) $|0\rangle + |1\rangle$ never measures as $|0\rangle - |1\rangle$ (due to "interference"), but in the state $|00\rangle + |11\rangle$, the first particle will measure as $|0\rangle - |1\rangle$ half the time. This is the same thing we would get if we didn't know $\theta$ in $|0\rangle + e^{i\theta}|1\rangle$, but that doesn't mean that anything actually changed a phase anywhere, right?
– A_P
Commented Dec 4, 2022 at 0:03
• @A_P 1. Generally, a state evolves through Hilbert space, and for a fixed basis almost every point in Hilbert space must be written as a linear combination of multiple basis states. 2. Loss of coherence comes from loss of (or change in the state of) information, by coupling to the environment or other systems (even by e.g. measurement). In your case of the Bell state, both states are needed to characterize the system. You're right that the reduced density matrix in that case IS the same as if you did not know the phase: what's important is that the model captures your lack of information. Commented Dec 5, 2022 at 21:20

Coherence is a term coming from studying waves with their wave equations. The wave equations have sinusoidal solutions , that have the frequency of the wave modeled. These solutions for the same frequency can have fixed phases between them .

coherence, a fixed relationship between the phase of waves in a beam of radiation of a single frequency. Two beams of light are coherent when the phase difference between their waves is constant; they are noncoherent if there is a random or changing phase relationship

In quantum mechanics the wave equation is related with the probability distribution, and quantum coherence of two wave functions means that the probability distribution will show the interference patterns in a superposition of states.

To get a feeling see this one electron at a time scattering on double slits. The same momentum of electrons impinges on the screen (same frequency) and the coherence is seen in the interference pattern of the probability distribution.

• Thank you! Can you recommend any definition of coherence that I could cite in a book? Commented May 3, 2022 at 13:09